Fifth seed Ruud saved two set points in the third set on his way to a 6-1 6-4 7-6 (4) victory over Italian Berrettini under the roof on Arthur Ashe Stadium.
The French Open finalist must make the final to have a chance of pipping Rafael Nadal to top spot in the rankings and if he were to meet Carlos Alcaraz for the trophy, the winner would also become world number one.
The Norwegian is certainly well in the mix to win his first grand slam title, with defeats for Nadal, Daniil Medvedev and Marin Cilic in the fourth round meaning there will be a new men’s major champion.
Ruud said of the ranking: “I don’t want to think too much about it. It’s of course something that all young players dream about. Let’s see if I can accomplish it.
“Of course it’s a little bit of extra motivation to dig in. If I’m very, very lucky, I can leave New York as world number one.”
Ruud initially found success on clay but has turned himself into an all-court player, with his powerful forehand, consistency and athleticism his main strengths.
He dominated Berrettini in the early stages, sweeping into a 6-1 5-1 lead before the Italian rallied and made a fight of the second set.
Berrettini then led 5-2 in the third and had two set points but Ruud hit back and dominated the tie-break to book his place in his second slam semi-final.
The 23-year-old said: “It was a better start than I ever had before in a match, everything was going my way. I was able to take care of the chances that I got.
“I got a little bit nervous towards the end of the second set because things were almost going too well. Sometimes you can get a little bit excited and think that you can walk on water. I managed to serve out the second set and the third was very tough.”
Berrettini was disappointed with his performance, saying: “I think he had the best start possible, and I helped him. He played a really good match, and I played a really bad match.
“Really nothing I can say more than the worst day of the tournament probably in the most important moment. I fought through but it wasn’t enough. I wasn’t feeling my game. I wasn’t feeling my mindset. I think he was feeling really good.”